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PHYSICAL REVIEW E

VOLUME 56, NUMBER 4

OCTOBER 1997

Real-time experimental control of a system in its chaotic and nonchaotic regimes David J. Christini,1 Visarath In,2 Mark L. Spano,2 William L. Ditto,3 and James J. Collins1 1

Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, Massachusetts 02215 2 Naval Surface Warfare Center, Carderock Division, West Bethesda, Maryland 20817 3 Applied Chaos Laboratory, School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 ~Received 1 July 1997! Current model-independent control techniques are limited, from a practical standpoint, by their dependence on a precontrol learning stage. Here we use a real-time, adaptive, model-independent ~RTAMI! feedback control technique to control an experimental system — a driven magnetoelastic ribbon — in its nonchaotic and chaotic regimes. We show that the RTAMI technique is capable of tracking and stabilizing higher-order unstable periodic orbits. These results demonstrate that the RTAMI technique is practical for on-the-fly ~i.e., no learning stage! control of real-world dynamical systems. @S1063-651X~97!50710-0# PACS number~s!: 05.45.1b, 75.80.1q

Model-independent chaos control techniques, the first of which was developed by Ott, Grebogi, and Yorke @1#, have been applied to a wide range of physical and physiological systems @2–11#. Recently, similar techniques have been developed to stabilize underlying unstable periodic orbits ~UPO’s! in nonchaotic dynamical systems @12–18#. In general, model-independent control techniques use feedback perturbations to stabilize a dynamical system about one of its UPO’s. In contrast to traditional control techniques ~which require knowledge of a system’s governing equations!, model-independent techniques are inherently well-suited for ‘‘black-box’’ systems because they extract all necessary control information from a premeasured time series. The flexibility of model independence in current dynamical control techniques, however, does not come without limitations. The precontrol time-series measurement and the corresponding system-dynamics estimation comprise a ‘‘learning’’ stage. For some real-world systems ~e.g., cardiac arrhythmias!, however, unwanted dynamics must be eliminated quickly, and thus the time required for a learning stage may be unavailable. Recently, a real-time, adaptive, model-independent ~RTAMI! control technique, was developed @19# to stabilize flip-saddle UPO’s in chaotic and nonchaotic dynamical systems that can be described effectively by a unimodal onedimensional map. Because the RTAMI technique does not require a precontrol learning stage ~i.e., it operates in real time! it is practical for on-the-fly control of dynamical systems. In Ref. @19#, the RTAMI technique was successfully applied to a wide range of model systems in their nonchaotic and chaotic regimes. Here, we apply the RTAMI control technique to an experimental system — a driven magnetoelastic ribbon — in its nonchaotic and chaotic regimes. The RTAMI technique is designed to stabilize the flipsaddle unstable periodic fixed point j*5 @ x * ,x * # T ~where superscript T denotes transpose and @ x * ,x * # T is a 231 column vector! of a system that can be described effectively by a unimodal one-dimensional map x n11 5 f (x n , p n ), where x n is the current value ~scalar! of one measurable system variable, x n11 is the next value of the same variable, and p n is the value ~scalar! of an accessible system parameter p at index n. The control technique perturbs p such that p n 5 ¯ p 1063-651X/97/56~4!/3749~4!/$10.00

56

1dpn , where ¯ p is the nominal parameter value, and d p n is a perturbation @3,4,20–22# given by

d p n5

x n 2x * n gn

,

~1!

where x * n is the current estimate of x * , and g n is the control sensitivity g at index n. The ideal value of g is the sensitivity of x * to perturbations: g ideal5 d x * / d p. As described in Ref. @23#, control can be achieved for nonideal values of g in the range u g u min< u g u < u g u max . ~Prior to control, it is not possible to determine g min or g max without an analytical system model or a learning stage.! As shown in Fig. 1, the current state point jn would move,

FIG. 1. First-return map showing that d p n @Eq. ~1!, with g 5g ideal# shifts the map from f (x n ,p n ) to f (x n ,p n 1 d p n ) such that 8 ' j* , rather than to its the next system state point is forced to jn11 expected position jˆ n11 . These data, shown for illustrative purposes, are from simulations of the Belousov-Zhabotinsky chemical reaction. R3749

© 1997 The American Physical Society

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in the absence of a perturbation ~i.e., d p n 50!, to jˆ n11 ~via the dotted arrow!. However, the control perturbation of Eq. ~1! ~corresponding to g5g ideal) shifts f (x n , p n ) to f (x n ,p n 8 5x * , instead of xˆ n11 . On 1 d p n ) such that x n maps to x n11 the first-return map, this shift appears as the movement of jn to j8n ~via the solid vertical arrow in Fig. 1!. When the map is returned to f (x n ,p n ) for the next iteration, the next state 8 ' j* , as desired for control. In a physical point will be jn11 system, due to noise, measurement errors, and the instability of j* , perturbations are required at each iteration to hold jn within the neighborhood of j* . Learning-stage dependent techniques use static values for x * and/or g, as estimated from a precontrol time-series measurement. In contrast, the RTAMI technique repeatedly estimates x * and g. In addition to eliminating the need for a learning stage, this adaptability allows for the control of nonstationary systems. When control is initiated, g can be set to an arbitrary value ~with the restriction that the sign of g must match that of g ideal ; if the signs do not match, control will fail!. After each measurement of x n , x * is estimated using N21

x* n5

( i50

x n2i , N

~2!

where N is the number of past data points included in the average @24#. Equation ~2! converges to x * because consecutive x n alternate on either side of x * due to the flip-saddle nature of j* . At each iteration, after x * is re-estimated via Eq. ~2!, the RTAMI technique evaluates whether the estimate of g should be adapted. The value of g is not adapted if the desired control precision e has been achieved. Control precision has not been achieved if

* u.e u x n 2x n21

~3!

is satisfied by at least L data points out of the N previous * is the estimate of x * that was tardata points, where x n21 geted for a given x n . The L/N factor is used @instead of a single evaluation of Eq. ~3!# to reduce the influence of noise and spurious data points. If the desired control precision has not been achieved @i.e., Eq. ~3! has been satisfied by at least L data points out of the N previous data points#, then the magnitude of g is adapted in accordance with the expected perturbation dynamics @19#. If g5g ideal , then the perturbation moves the state point from its current position jn to j* ~as in Fig. 1!. If u g u is too large ~i.e., d p is too small!, then the state point moves from its current position jn to a position closer to j* than would be expected without a perturbation. If u g u is too small ~i.e., d p is too large!, then the state point moves from its current position jn to a position on the same side of the line of identity. ~This is in contrast to the expected alternation, due to the flip-saddle nature of j* , of consecutive state points on either side of the line of identity.! The criterion sgn~ x n 2x n21 ! 5sgn~ x n21 2x n22 !

~4!

is satisfied when two consecutive state points ( @ x n21 ,x n22 # and @ x n ,x n21 # ) lie on the same side of the line of identity. The RTAMI technique increases the magnitude of g ~i.e.,

FIG. 2. ~a! x n , ~b! H dcn , and ~c! g n versus drive cycle n for a RTAMI control trial of the chaotic magnetoelastic ribbon. The respective control stages are annotated in ~a!, ~b!, and ~c!.

g n11 5g n r , where r is the adjustment factor! if Eq. ~4! is satisfied for at least L data points out of the N previous data points. As with the evaluation of control precision @Eq. ~3!#, the L/N factor is used @instead of a single evaluation of Eq. ~4!# to reduce the influence of noise and spurious data points. If the magnitude of g is not increased @as dictated by Eq. ~4!#, then the magnitude of g is decreased if jn is not converging rapidly ~at a rate governed by r) to j* . Specifically, the magnitude of g is decreased ~i.e., g n11 5g n / r ) if N21

* * u x n2i21 2x n2i22 u 2 u x n2i 2x n2i21 u 1 ,r% . N i50 * u x n2i21 2x n2i22 u

(

~5!

Equation ~5! is satisfied if, on average, the distance * u x n2i 2x n2i21 u between a given data point x n2i and its cor* is not at least r% responding fixed-point estimate x n2i21 * u between the presmaller than the distance u x n2i21 2x n2i22 vious data point x n2i21 and the previous fixed-point estimate * x n2i22 . If neither Eq. ~4! nor Eq. ~5! is satisfied, then g is not adapted because x is properly approaching the estimate of x *. The experimental system we considered @25# consists of a gravitationally buckled magnetoelastic ribbon driven parametrically by a sinusoidally varying magnetic field. The ribbon is clamped at its lower end and its position x is measured once per drive period at a point a short distance above the clamp. The ribbon’s Young’s modulus can be varied by applying an external magnetic field. The applied magnetic field

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56

REAL-TIME EXPERIMENTAL CONTROL OF A SYSTEM . . .

R3751

FIG. 4. ~a! x versus H dc for a RTAMI tracking trial ~dark points! overlaid onto the corresponding bifurcation diagram. ~b! g for the tracking trial shown in ~a!. FIG. 3. ~a! x n , ~b! H dcn , and ~c! g n versus drive cycle n for a RTAMI control trial of the magnetoelastic ribbon in two different nonchaotic regimes @stable period-4 regime (1

Lihat lebih banyak...
PHYSICAL REVIEW E

VOLUME 56, NUMBER 4

OCTOBER 1997

Real-time experimental control of a system in its chaotic and nonchaotic regimes David J. Christini,1 Visarath In,2 Mark L. Spano,2 William L. Ditto,3 and James J. Collins1 1

Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, Massachusetts 02215 2 Naval Surface Warfare Center, Carderock Division, West Bethesda, Maryland 20817 3 Applied Chaos Laboratory, School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 ~Received 1 July 1997! Current model-independent control techniques are limited, from a practical standpoint, by their dependence on a precontrol learning stage. Here we use a real-time, adaptive, model-independent ~RTAMI! feedback control technique to control an experimental system — a driven magnetoelastic ribbon — in its nonchaotic and chaotic regimes. We show that the RTAMI technique is capable of tracking and stabilizing higher-order unstable periodic orbits. These results demonstrate that the RTAMI technique is practical for on-the-fly ~i.e., no learning stage! control of real-world dynamical systems. @S1063-651X~97!50710-0# PACS number~s!: 05.45.1b, 75.80.1q

Model-independent chaos control techniques, the first of which was developed by Ott, Grebogi, and Yorke @1#, have been applied to a wide range of physical and physiological systems @2–11#. Recently, similar techniques have been developed to stabilize underlying unstable periodic orbits ~UPO’s! in nonchaotic dynamical systems @12–18#. In general, model-independent control techniques use feedback perturbations to stabilize a dynamical system about one of its UPO’s. In contrast to traditional control techniques ~which require knowledge of a system’s governing equations!, model-independent techniques are inherently well-suited for ‘‘black-box’’ systems because they extract all necessary control information from a premeasured time series. The flexibility of model independence in current dynamical control techniques, however, does not come without limitations. The precontrol time-series measurement and the corresponding system-dynamics estimation comprise a ‘‘learning’’ stage. For some real-world systems ~e.g., cardiac arrhythmias!, however, unwanted dynamics must be eliminated quickly, and thus the time required for a learning stage may be unavailable. Recently, a real-time, adaptive, model-independent ~RTAMI! control technique, was developed @19# to stabilize flip-saddle UPO’s in chaotic and nonchaotic dynamical systems that can be described effectively by a unimodal onedimensional map. Because the RTAMI technique does not require a precontrol learning stage ~i.e., it operates in real time! it is practical for on-the-fly control of dynamical systems. In Ref. @19#, the RTAMI technique was successfully applied to a wide range of model systems in their nonchaotic and chaotic regimes. Here, we apply the RTAMI control technique to an experimental system — a driven magnetoelastic ribbon — in its nonchaotic and chaotic regimes. The RTAMI technique is designed to stabilize the flipsaddle unstable periodic fixed point j*5 @ x * ,x * # T ~where superscript T denotes transpose and @ x * ,x * # T is a 231 column vector! of a system that can be described effectively by a unimodal one-dimensional map x n11 5 f (x n , p n ), where x n is the current value ~scalar! of one measurable system variable, x n11 is the next value of the same variable, and p n is the value ~scalar! of an accessible system parameter p at index n. The control technique perturbs p such that p n 5 ¯ p 1063-651X/97/56~4!/3749~4!/$10.00

56

1dpn , where ¯ p is the nominal parameter value, and d p n is a perturbation @3,4,20–22# given by

d p n5

x n 2x * n gn

,

~1!

where x * n is the current estimate of x * , and g n is the control sensitivity g at index n. The ideal value of g is the sensitivity of x * to perturbations: g ideal5 d x * / d p. As described in Ref. @23#, control can be achieved for nonideal values of g in the range u g u min< u g u < u g u max . ~Prior to control, it is not possible to determine g min or g max without an analytical system model or a learning stage.! As shown in Fig. 1, the current state point jn would move,

FIG. 1. First-return map showing that d p n @Eq. ~1!, with g 5g ideal# shifts the map from f (x n ,p n ) to f (x n ,p n 1 d p n ) such that 8 ' j* , rather than to its the next system state point is forced to jn11 expected position jˆ n11 . These data, shown for illustrative purposes, are from simulations of the Belousov-Zhabotinsky chemical reaction. R3749

© 1997 The American Physical Society

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R3750

CHRISTINI, IN, SPANO, DITTO, AND COLLINS

56

in the absence of a perturbation ~i.e., d p n 50!, to jˆ n11 ~via the dotted arrow!. However, the control perturbation of Eq. ~1! ~corresponding to g5g ideal) shifts f (x n , p n ) to f (x n ,p n 8 5x * , instead of xˆ n11 . On 1 d p n ) such that x n maps to x n11 the first-return map, this shift appears as the movement of jn to j8n ~via the solid vertical arrow in Fig. 1!. When the map is returned to f (x n ,p n ) for the next iteration, the next state 8 ' j* , as desired for control. In a physical point will be jn11 system, due to noise, measurement errors, and the instability of j* , perturbations are required at each iteration to hold jn within the neighborhood of j* . Learning-stage dependent techniques use static values for x * and/or g, as estimated from a precontrol time-series measurement. In contrast, the RTAMI technique repeatedly estimates x * and g. In addition to eliminating the need for a learning stage, this adaptability allows for the control of nonstationary systems. When control is initiated, g can be set to an arbitrary value ~with the restriction that the sign of g must match that of g ideal ; if the signs do not match, control will fail!. After each measurement of x n , x * is estimated using N21

x* n5

( i50

x n2i , N

~2!

where N is the number of past data points included in the average @24#. Equation ~2! converges to x * because consecutive x n alternate on either side of x * due to the flip-saddle nature of j* . At each iteration, after x * is re-estimated via Eq. ~2!, the RTAMI technique evaluates whether the estimate of g should be adapted. The value of g is not adapted if the desired control precision e has been achieved. Control precision has not been achieved if

* u.e u x n 2x n21

~3!

is satisfied by at least L data points out of the N previous * is the estimate of x * that was tardata points, where x n21 geted for a given x n . The L/N factor is used @instead of a single evaluation of Eq. ~3!# to reduce the influence of noise and spurious data points. If the desired control precision has not been achieved @i.e., Eq. ~3! has been satisfied by at least L data points out of the N previous data points#, then the magnitude of g is adapted in accordance with the expected perturbation dynamics @19#. If g5g ideal , then the perturbation moves the state point from its current position jn to j* ~as in Fig. 1!. If u g u is too large ~i.e., d p is too small!, then the state point moves from its current position jn to a position closer to j* than would be expected without a perturbation. If u g u is too small ~i.e., d p is too large!, then the state point moves from its current position jn to a position on the same side of the line of identity. ~This is in contrast to the expected alternation, due to the flip-saddle nature of j* , of consecutive state points on either side of the line of identity.! The criterion sgn~ x n 2x n21 ! 5sgn~ x n21 2x n22 !

~4!

is satisfied when two consecutive state points ( @ x n21 ,x n22 # and @ x n ,x n21 # ) lie on the same side of the line of identity. The RTAMI technique increases the magnitude of g ~i.e.,

FIG. 2. ~a! x n , ~b! H dcn , and ~c! g n versus drive cycle n for a RTAMI control trial of the chaotic magnetoelastic ribbon. The respective control stages are annotated in ~a!, ~b!, and ~c!.

g n11 5g n r , where r is the adjustment factor! if Eq. ~4! is satisfied for at least L data points out of the N previous data points. As with the evaluation of control precision @Eq. ~3!#, the L/N factor is used @instead of a single evaluation of Eq. ~4!# to reduce the influence of noise and spurious data points. If the magnitude of g is not increased @as dictated by Eq. ~4!#, then the magnitude of g is decreased if jn is not converging rapidly ~at a rate governed by r) to j* . Specifically, the magnitude of g is decreased ~i.e., g n11 5g n / r ) if N21

* * u x n2i21 2x n2i22 u 2 u x n2i 2x n2i21 u 1 ,r% . N i50 * u x n2i21 2x n2i22 u

(

~5!

Equation ~5! is satisfied if, on average, the distance * u x n2i 2x n2i21 u between a given data point x n2i and its cor* is not at least r% responding fixed-point estimate x n2i21 * u between the presmaller than the distance u x n2i21 2x n2i22 vious data point x n2i21 and the previous fixed-point estimate * x n2i22 . If neither Eq. ~4! nor Eq. ~5! is satisfied, then g is not adapted because x is properly approaching the estimate of x *. The experimental system we considered @25# consists of a gravitationally buckled magnetoelastic ribbon driven parametrically by a sinusoidally varying magnetic field. The ribbon is clamped at its lower end and its position x is measured once per drive period at a point a short distance above the clamp. The ribbon’s Young’s modulus can be varied by applying an external magnetic field. The applied magnetic field

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REAL-TIME EXPERIMENTAL CONTROL OF A SYSTEM . . .

R3751

FIG. 4. ~a! x versus H dc for a RTAMI tracking trial ~dark points! overlaid onto the corresponding bifurcation diagram. ~b! g for the tracking trial shown in ~a!. FIG. 3. ~a! x n , ~b! H dcn , and ~c! g n versus drive cycle n for a RTAMI control trial of the magnetoelastic ribbon in two different nonchaotic regimes @stable period-4 regime (1

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